The homogenization method of roughness analysis in turbulent lubrication

The hydrodynamic lubrication theory founded by Osborne Reynolds in 1886 is based on several assumptions, including that the contact surfaces are perfectly smooth, and the lubricant flow regime is laminar. However, in certain special cases these hypotheses may not be valid, and it is seen that surface roughness and turbulence affect bearing performance. The approach adopted in the present work is based on the application of the homogenization method to the turbulent Reynolds equation. In homogenization, multiscale expansion of the fluid pressure terms leads to a system of four partial differential equations governing two types of problems (local and global). The solutions are, respectively, periodic functions, and the homogenized pressure. As an illustrative example, a journal bearing with a rough surface operating in the turbulent regime is analyzed. Numerical simulations are performed by imposing periodic isotropic roughness patterns over the stationary outer cylinder surface. The homogenized solutions are compared to direct solutions of the deterministic problem. Homogenization is shown to be a powerful but practical method to attack problems of rough surfaces and is applied to turbulent lubrication for the first time. In the plain journal bearing example, the effect of roughness is to increase the load significantly in many cases. The effect of turbulence is to greatly increase the load, in both the rough and smooth cases.